0=x^2+6x+6

Solution for 0=x^2+6x+6 equation:

Simplifying
0 = x2 + 6x + 6

Reorder the terms:
0 = 6 + 6x + x2

Solving
0 = 6 + 6x + x2

Solving for variable 'x'.

Combine like terms: 0 + -6 = -6
-6 + -6x + -1x2 = 6 + 6x + x2 + -6 + -6x + -1x2

Reorder the terms:
-6 + -6x + -1x2 = 6 + -6 + 6x + -6x + x2 + -1x2

Combine like terms: 6 + -6 = 0
-6 + -6x + -1x2 = 0 + 6x + -6x + x2 + -1x2
-6 + -6x + -1x2 = 6x + -6x + x2 + -1x2

Combine like terms: 6x + -6x = 0
-6 + -6x + -1x2 = 0 + x2 + -1x2
-6 + -6x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-6 + -6x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(6 + 6x + x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(6 + 6x + x2)' equal to zero and attempt to solve:

Simplifying
6 + 6x + x2 = 0

Solving
6 + 6x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 6x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + 6x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + 6x + x2 = 0 + -6
6x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
6x + x2 = -6

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -6 + 9

Reorder the terms:
9 + 6x + x2 = -6 + 9

Combine like terms: -6 + 9 = 3
9 + 6x + x2 = 3

Factor a perfect square on the left side:
(x + 3)(x + 3) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(x + 3) equal to 1.732050808 and -1.732050808.

Subproblem 1
x + 3 = 1.732050808

Simplifying
x + 3 = 1.732050808

Reorder the terms:
3 + x = 1.732050808

Solving
3 + x = 1.732050808

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + x = 1.732050808 + -3

Combine like terms: 3 + -3 = 0
0 + x = 1.732050808 + -3
x = 1.732050808 + -3

Combine like terms: 1.732050808 + -3 = -1.267949192
x = -1.267949192

Simplifying
x = -1.267949192

Subproblem 2
x + 3 = -1.732050808

Simplifying
x + 3 = -1.732050808

Reorder the terms:
3 + x = -1.732050808

Solving
3 + x = -1.732050808

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + x = -1.732050808 + -3

Combine like terms: 3 + -3 = 0
0 + x = -1.732050808 + -3
x = -1.732050808 + -3

Combine like terms: -1.732050808 + -3 = -4.732050808
x = -4.732050808

Simplifying
x = -4.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.267949192, -4.732050808}
Solution
x = {-1.267949192, -4.732050808}